A generalized Dupire formula and a stable way to estimate it P. - PowerPoint PPT Presentation

A generalized Dupire formula and a stable way to estimate it P. Mayer mayer@opt.math.tugraz.at Institute for Mathematics Graz University of Technology RICAM Special Semester Concluding Workshop Linz, 2.12.2008 Based on joint work with D. Belomestny

Outline Introduction On the way to a formula Robust estimation 2.12.2008 P.Mayer 2 / 21

Outline Introduction On the way to a formula Robust estimation 2.12.2008 P.Mayer Introduction 3 / 21

The well known Dupire model. We all know: B-S-M model not capable of explaining vol smile. ⇒ Dupire, Derman & Kani ”engineered” market model, that is: S T = e ( r − η ) T e X T , with � t � t σ 2 ( s , S s ) X t = − ds + σ ( s , S s ) dW s . 2 0 0 Other way to understand the above: � t � t σ 2 ( s , S s ) ds + ˜ σ 2 ( s , S s ) ds ) X t = − W ( 2 0 0 � t σ 2 ( s , S s ) ds ) , = Y ( 0 where Y ( t ) = ˜ W ( t ) − t / 2. 2.12.2008 P.Mayer Introduction 4 / 21

The well known Dupire model. We all know: B-S-M model not capable of explaining vol smile. ⇒ Dupire, Derman & Kani ”engineered” market model, that is: S T = e ( r − η ) T e X T , with � t � t σ 2 ( s , S s ) X t = − ds + σ ( s , S s ) dW s . 2 0 0 Other way to understand the above: � t � t σ 2 ( s , S s ) ds + ˜ σ 2 ( s , S s ) ds ) X t = − W ( 2 0 0 � t σ 2 ( s , S s ) ds ) , = Y ( 0 where Y ( t ) = ˜ W ( t ) − t / 2. 2.12.2008 P.Mayer Introduction 4 / 21

Drawbacks of the local volatility model. √ often very steep volatility structure √ Problems when pricing path-dependent options √ No jumps possible Possible improvement: L´ evy models Advantages: √ jump-risk included √ fat tails √ skewed log-returns via asymmetric L´ evy measure 2.12.2008 P.Mayer Introduction 5 / 21

Drawbacks of the local volatility model. √ often very steep volatility structure √ Problems when pricing path-dependent options √ No jumps possible Possible improvement: L´ evy models Advantages: √ jump-risk included √ fat tails √ skewed log-returns via asymmetric L´ evy measure 2.12.2008 P.Mayer Introduction 5 / 21

Why taking Brownian motion? Carr et al.: local L´ evy market model : S T = e ( r − η ) T e X T . X time-changed L´ evy process: X t = Y ( A ( t )) , � t where A ( T ) = 0 a 0 ( S t − , t ) dt and Y given L´ evy process. Assumptions: √ ∃ δ > 0 s.t. L´ R ( y e y (1+ δ ) − y ) ν ( dy ) < ∞ . � evy measure ν of Y fulfills √ 0 < a ≤ a 0 ≤ a < ∞ such that SDE solvable & pdf of S continuous. Question: How to identify local speed function a 0 ? 2.12.2008 P.Mayer Introduction 6 / 21

Why taking Brownian motion? Carr et al.: local L´ evy market model : S T = e ( r − η ) T e X T . X time-changed L´ evy process: X t = Y ( A ( t )) , � t where A ( T ) = 0 a 0 ( S t − , t ) dt and Y given L´ evy process. Assumptions: √ ∃ δ > 0 s.t. L´ R ( y e y (1+ δ ) − y ) ν ( dy ) < ∞ . � evy measure ν of Y fulfills √ 0 < a ≤ a 0 ≤ a < ∞ such that SDE solvable & pdf of S continuous. Question: How to identify local speed function a 0 ? 2.12.2008 P.Mayer Introduction 6 / 21

What is the task? √ Wanted: local speed function a 0 . √ European options liquid ⇒ Usable for calibration. � e − rT (e ( r − η ) T e Y ( A ( T )) − K ) + � C ( a 0 ) ( K , T ) := E Q Forward: a 0 → Model price of European option C M ( K , T ) Inverse: ˆ a 0 ← Market price of European option Problems: √ Existence of ˆ a 0 ? √ Stability of ˆ a 0 ? √ Computationally feasible algorithm to find ˆ a 0 ? 2.12.2008 P.Mayer Introduction 7 / 21

What is the task? √ Wanted: local speed function a 0 . √ European options liquid ⇒ Usable for calibration. � e − rT (e ( r − η ) T e Y ( A ( T )) − K ) + � C ( a 0 ) ( K , T ) := E Q Forward: a 0 → Model price of European option C M ( K , T ) Inverse: ˆ a 0 ← Market price of European option Problems: √ Existence of ˆ a 0 ? √ Stability of ˆ a 0 ? √ Computationally feasible algorithm to find ˆ a 0 ? 2.12.2008 P.Mayer Introduction 7 / 21

What is the task? √ Wanted: local speed function a 0 . √ European options liquid ⇒ Usable for calibration. � e − rT (e ( r − η ) T e Y ( A ( T )) − K ) + � C ( a 0 ) ( K , T ) := E Q Forward: a 0 → Model price of European option C M ( K , T ) Inverse: ˆ a 0 ← Market price of European option Problems: √ Existence of ˆ a 0 ? √ Stability of ˆ a 0 ? √ Computationally feasible algorithm to find ˆ a 0 ? 2.12.2008 P.Mayer Introduction 7 / 21

Outline Introduction On the way to a formula Robust estimation 2.12.2008 P.Mayer On the way to a formula 8 / 21

Getting lsf and call prices in touch. Using the Tanaka-Meyer formula and taking expectation: �� T � e rT C( K , T ) = ( S 0 − K ) + + E ( r − η ) S t − 1 { S t − > K } dt 0 �� T a 0 ( S t − , t ) σ 2 � 2 δ ( S t − − K ) S 2 + E t − dt 0 �� T � log( K � St − ) ( K − S t − e x ) ν ( dx ) dt + E 1 { S t − > K } a 0 ( S t − , t ) 0 −∞   � T � ∞ ( S t − e x − K ) ν ( dx ) dt  . + E 1 { S t − ≤ K } a 0 ( S t − , t )  K 0 log( St − ) ⇒ implicit equation for a 0 . 2.12.2008 P.Mayer On the way to a formula 9 / 21

Can we get a more explicit relation? Define √ γ ( k , T ) = e η T C(e k +( r − η ) T , T ) √ a ( y , T ) = a 0 (e y +( r − η ) T , T ) √ ψ . . . double exponential tail of L´ evy measure  R z −∞ ( e z − e x ) ν ( dx ) for z < 0 ψ ( z ) = R ∞ ( e x − e z ) ν ( dx ) for z > 0 . z Then ...after some calculations...: F ( γ T ( ., T )) � � F ( ψ ) + σ 2 = F ( γ kk ( ., T ) − γ k ( ., T )) a ( ., T ) . cf. Carr et al. (2004) and Dupire formula: F − 1 ( 1 F ( ψ )+ σ 2 ) ∗ γ T a = . γ kk − γ k 2.12.2008 P.Mayer On the way to a formula 10 / 21

Can we get a more explicit relation? Define √ γ ( k , T ) = e η T C(e k +( r − η ) T , T ) √ a ( y , T ) = a 0 (e y +( r − η ) T , T ) √ ψ . . . double exponential tail of L´ evy measure  R z −∞ ( e z − e x ) ν ( dx ) for z < 0 ψ ( z ) = R ∞ ( e x − e z ) ν ( dx ) for z > 0 . z Then ...after some calculations...: F ( γ T ( ., T )) � � F ( ψ ) + σ 2 = F ( γ kk ( ., T ) − γ k ( ., T )) a ( ., T ) . cf. Carr et al. (2004) and Dupire formula: F − 1 ( 1 F ( ψ )+ σ 2 ) ∗ γ T a = . γ kk − γ k 2.12.2008 P.Mayer On the way to a formula 10 / 21

Can we get a more explicit relation? Define √ γ ( k , T ) = e η T C(e k +( r − η ) T , T ) √ a ( y , T ) = a 0 (e y +( r − η ) T , T ) √ ψ . . . double exponential tail of L´ evy measure  R z −∞ ( e z − e x ) ν ( dx ) for z < 0 ψ ( z ) = R ∞ ( e x − e z ) ν ( dx ) for z > 0 . z Then ...after some calculations...: F ( γ T ( ., T )) � � F ( ψ ) + σ 2 = F ( γ kk ( ., T ) − γ k ( ., T )) a ( ., T ) . cf. Carr et al. (2004) and Dupire formula: F − 1 ( 1 F ( ψ )+ σ 2 ) ∗ γ T a = . γ kk − γ k 2.12.2008 P.Mayer On the way to a formula 10 / 21

How does F ( ψ ) look like? Denoting � ∞ e y i ω − 1 − y i ω ˆ � � k ( ω ) = ν ( dy ) , −∞ pure-jump term part in the L´ evy-Khintchine formula associated to the L´ evy density ν with the identity as ”‘truncation function”’. Fourier transform of ψ : k ( − i − ω ) + (i ω − 1)ˆ ˆ k ( − i) F ( ψ )( ω ) = √ . 2 π i ω (i ω − 1) 2.12.2008 P.Mayer On the way to a formula 11 / 21

How does F ( ψ ) look like? Denoting � ∞ e y i ω − 1 − y i ω ˆ � � k ( ω ) = ν ( dy ) , −∞ pure-jump term part in the L´ evy-Khintchine formula associated to the L´ evy density ν with the identity as ”‘truncation function”’. Fourier transform of ψ : k ( − i − ω ) + (i ω − 1)ˆ ˆ k ( − i) F ( ψ )( ω ) = √ . 2 π i ω (i ω − 1) 2.12.2008 P.Mayer On the way to a formula 11 / 21

Some facts for pure jump processes. For pure jump-processes: √ F ( ψ )( ω ) � = 0 so division justified. √ for the asymptotic Proposition The asymptotic of the Fourier transformed double exponential tail can be calculated as follows: � � 1 � | ω | min(1 , 2 − β ) � � � � = O for | ω | → ∞ , � � F ( ψ )( ω ) � where β = sup { β : | x | − β = o ( ν ( R / [ − x , x ])) for x → 0 } . 2.12.2008 P.Mayer On the way to a formula 12 / 21

An example Y tempered stable process � e − λ + x for x > 0 x 1+ α + ν ( x ) = e λ − x for x < 0 . | x | 1+ α − Then for α ± � = 0 , 1 and for simplicity λ = λ − = λ + and α = α + = α − k ( ω ) = 2 λ α Γ( − α ) ((1 − i ω/λ ) α − 1 + α i ω/λ ) . ˆ and k ( − i − ω ) + (i ω − 1)ˆ ˆ k ( − i) F ( ψ )( ω ) = √ 2 π i ω (i ω − 1) 2 λ α Γ( − α ) n (1 − 1 /λ + i ω/λ ) α − (1 − 1 /λ ) α − i ω (1 − (1 − 1 /λ ) α ) o = √ . 2 π i ω (i ω − 1) Hence: 1 / F ( ψ )( ω ) ∼ 1 for ω → 0, | 1 / F ( ψ )( ω ) | ∼ | ω | min(2 − α, 1) for | ω | → ∞ 2.12.2008 P.Mayer On the way to a formula 13 / 21

Outline Introduction On the way to a formula Robust estimation 2.12.2008 P.Mayer Robust estimation 14 / 21